# 4.2.18. FFPT¶

The program FFPT prepares the one-electron integral file generated by SEWARD for subsequent finite-field perturbation calculations. To do so, the core Hamiltonian matrix is always reconstructed from the nuclear attraction and kinetic energy integrals. The perturbation matrix is then added to the core Hamiltonian matrix where the external perturbation and its strength is specified by input. Any suitable combination of the perturbations is allowed. Following some examples

**Dipole moment operator:**This option corresponds to a homogeneous external field perturbation and can be used to calculate dipole moments and dipole polarizabilities.**Quadrupole and higher electric moment operators:**This option corresponds to a non homogeneous external field perturbation and can be used to calculate quadrupole moments and quadrupole polarizabilities, etc.**Relativistic corrections:**This option is used to calculate perturbational relativistic corrections (sum of the mass-velocity and the one-electron Darwin contact term) to the total energy. Note that care must be taken to avoid variational collapse, i.e. the perturbation correction should be small.

For a complete list of one-electron integrals which can be evaluated by the program SEWARD check out the section 4.2.48.2.1 and, especially, the subsection 5.2.1.3.

Note, the perturbation matrices consist of the electronic contributions, only. The quadrupole, electric field gradient and higher electric moment perturbation matrices are given as the traceless tensors.

## 4.2.18.1. Dependencies¶

In order to complete successfully, the program FFPT needs the one-electron integral file. The latter must include all types of integrals needed to construct the perturbed one-electron Hamiltonian.

## 4.2.18.3. Input¶

The input to the FFPT program begins with the program name:

```
&FFPT
```

### 4.2.18.3.1. General keywords¶

The following keywords are known to the FFPT utility:

- TITLe
- Followed by a title line
- DIPO
- Add the dipole moment perturbation operator. By default, the dipole moment integrals are always computed with respect to the center of nuclear charge. The keyword is followed by up to three additional input lines. Each line consists of two entries, the component of the dipole operator and the perturbation length. The component is specified by a single letter (X, Y or Z).
- QUAD
- Add the quadrupole moment perturbation operator. The keyword is followed by at least one additional input line and may be complemented by as many additional lines as needed. Each line consists of two entries, the component of the operator and the perturbation strength. The component is specified by a pair of letters (XX, XY, XZ, YY, YZ or ZZ). By default, the quadrupole moment integrals are calculated with respect to the center of mass. For any other selection the origin of the perturbation operator also needs to be specified by entering a line starting with the string ORIG followed by the coordinates.
- OCTU
- Add the octupole moment perturbation operator. The keyword is followed by at least one additional input line and may be complemented by as many additional lines as needed. Each line consists of two entries, the component of the operator and the perturbation strength. The component is specified by a triple of letters (XXX, XXY, XXZ, XYY, XYZ, XZZ, YYY, YYZ, YZZ, or ZZZ). By default, the octupole moment integrals are calculated with respect to the center of mass. For any other selection the origin of the perturbation operator also needs to be specified by entering a line starting with the string ORIG followed by the coordinates.
- EFLD
- Add the electric field perturbation operator. The keyword is followed by at least two additional input lines and may be complemented by as many additional lines as needed. Each line consists of two entries, the component of the operator and the perturbation strength. The component is specified by a single letter (X, Y or Z). In addition, the origin of the perturbation operator also needs to be specified by entering a line starting with the string ORIG followed by the coordinates.
- EFGR
- Add the electric field gradient perturbation operator. The keyword is followed by at least one additional input line and may be complemented by as many additional lines as needed. Each line consists of two entries, the component of the operator and the perturbation strength. The component is specified by a pair of letters (XX, XY, XZ, YY, YZ or ZZ). In addition, the origin of the perturbation operator also needs to be specified by entering a line starting with the string ORIG followed by the coordinates.
- RELA
- Add the relativistic correction (mass-velocity and one-electron Darwin contact term). The command is followed by one additional line of input specifying the perturbation strength.
- GLBL
This command marks the beginning of a more general perturbation description which is not included as a subcommand of the FFPT command. This card is followed by as many additional input lines as needed and is terminated if the next input line starts with a command. Each input line contains only one perturbation description and three data fields which are: Label, component and perturbation strength. The label consists of a character string of length 8 and names the one-electron integrals produced by SEWARD. The component of an operator is given as an integer. The last parameter denotes the strength of a perturbation operator and is given as a real number. For a list of the available one-electron integral labels refer to section 4.2.48.

For example to add Pauli repulsion integrals for reaction field calculations the input would look like:

&FFPT GLBL 'Well 1' 1 1.000 'Well 2' 1 1.000 'Well 3' 1 1.000

- SELEctive
With the same localization scheme as used in LOPROP, the perturbation from FFPT is localized in an orthogonal basis. Then the user can specify on which basis functions the perturbation should act. For example, the input

&FFPT DIPO X 0.005 SELECTIVE 2 .true. 1 26 .false. 67 82 .true. 0.5

leads to that the perturbation only acts on densities with (1) both basis function indexes in the set \(\{1,\ldots,26\}\) or (2) one index in the set \(\{1,\ldots,26\}\) while the other is in the set \(\{67,\ldots,82\}\), and in this case the perturbation should be multiplied by 0.5.; all other densities are unaffected by the perturbation. We call the former type of subset an atom domain and the latter a bond domain. Generally, the input structure is this: First line specifies how many subsets, \(N\), that will be defined. Then follow \(N\) lines starting with a logical flag telling if the subset is an atom domain with the starting and ending basis function indexes thereafter. \(N-1\) lines follow where the bond domain is defined in the following way:

Do i=2,nSets Read(*,*)(Bonds(i,j),j=1,i-1) Enddo

Finally a scalar is given which scales the defined bond domains.

The LoProp-functions will almost coincide with the original input AO-basis, although the localization will modify the meaning slightly, hence it is not possible to exactly localize the perturbation to a group of atoms; LOPROP is a way to come close to perfect localization. FFPT calls LOPROP internally and no call to LOPROP has to specified by the user.

- CUMUlative
- Adds the perturbation to the current H0, enabling many consecutive FFPT calls. Without this keyword, the perturbation always starts from the unperturbed H0.

## 4.2.18.4. Input example¶

The following input will prepare the one-electron integral file generated by SEWARD for subsequent finite-field perturbation calculations by adding a linear electric field in z-direction.

```
&FFPT
DIPO
Z 0.001
```

Response properties are obtained by numerical differentiation of the total energy with respect to the field parameter. For definitions of the response properties the interested reader is referred to the paper of A.D. Buckingham [69]. According to the definition of the dipole moment, it is obtained as the first derivative of the energy with respect to the field strength. Similarly, the dipole polarizability is given by the second derivative of the energy with respect to the field strength.